Can someone explain Backward Euler in the context of a transient thermal problem in 2-dimensions for simplicity, lets say I have a weak form of the thermal problem , i would like to understand how to move in time forward using backward euler and gets converted into an algebraic system ? or any good resource is also appreciated
Thanks in advance.
Consider the abstract evolution equation $\frac{du}{dt} = f(u).$ $f$ may contain nonlinearities and spatial differential operators. We now discretize the time derivative with a backwards difference: $\frac{du}{dt}(t_n) \approx \frac{u_n-u_{n-1}}{\Delta t}$, where $u_n$ is the approximation of $u(t_n)$. Our discrete evolution equation then becomes $$ \frac{u_n-u_{n-1}}{\Delta t} = f(u_n). $$ We can rearrange this into a nonlinear system so that $u_n$ is the solution to $F(v)=0$, where $F(v) = v - u_{n-1} - \Delta t f(v).$ If $f$ has sufficient regularity, one can show that this equation has a unique solution for sufficiently small $\Delta t$, leading to a well-posed iterative scheme. To actually solve the nonlinear system, we tend to use iterative schemes like the fixed point method or Newton’s method.
Like for many other things, deal.II has a tutorial program for that: https://dealii.org/developer/doxygen/deal.II/step_26.html
(Disclaimer: I'm one of the authors of deal.II.)
